13. Potential of Branching Processes as a Modeling Tool
217
tends to 1 − λ) and the stabilization conditional on nonextinction in the QSD. This
interplay between extinction and asymptotic stabilization conditional on nonextinction makes such a population behave asymptotically as a single individual whose survival probability from one time step to the next would be λ, the
asymptotic growth rate of the population. In the White Stork case, the high value
of 1 − λ tells us that asymptotically the population will become extinct relatively
quickly.
The aforementioned interplay between extinction and “quasi-stabilization” is
valid as soon as quasi-stationarity occurs, including for DD BPs. The above
parallel between deterministic models and BPs is, however, only valid in densityindependent, constant-environment situations, in which the expected population
sizes of the BPs are given by a Leslie matrix model. For DD BPs, we are not aware
of any simple deterministic model that would yield the successive expected population sizes of the BPs as functions of only the previous expected population size.
Indeed, the interplay between stochasticity and density dependence makes E (Z t )
depend not only on E(Z t−1 ) but on the whole probability distribution of Z t−1 .
Theoretical results also appear useful to frame simulations. We give three
illustrations of this statement:
1. Theoretical results emphasize certain quantities, with known asymptotic behavior, such as the expected population size conditional on nonextinction (e.g.,
Lebreton 1978; Gosselin 1996).
2. They lead to meaningful procedures for estimating some parameters. For example, once quasi-stationarity is reached, mean time to extinction is best
estimated by fitting a truncated geometric distribution to the empirical probability distribution of time to extinction. In simulations over a finite time interval, one may account in this way for the nonobserved tail of the probability
distribution of time to extinction and estimate properly the mean time to
extinction.
3. Last, theoretical results may help to distinguish the transient part of the simulation results (i.e., the part that is influenced by the initial conditions) from the
asymptotic one (e.g., Goodman 1987; Gabriel and B¨ urger 1992). The relative
role of the two components, and thus the relevance of the asymptotic results,
likely depend on a ratio
|λ 2 |
λ
, where λ, the asymptotic rate of growth, is the
largest eigenvalue, in modulus, of an equivalent of the matrix Q introduced in
Appendix 13.2. The second largest eigenvalue is λ 2 . If 1 −
|λ 2 |
λ
is much bigger
than 1 − λ, quasi-stationarity will be reached rapidly, before many extinctions
have occurred, in which case the results presented in this chapter are very
relevant. On the contrary, if 1 −
|λ 2 |
λ
is not much bigger than 1 − λ, many
extinctions are likely to take place before quasi-stationarity is reached. In this
last case, quasi-stationarity is less interesting (cf. Day and Possingham 1995;
Gosselin 1998b, c).
217
tends to 1 − λ) and the stabilization conditional on nonextinction in the QSD. This
interplay between extinction and asymptotic stabilization conditional on nonextinction makes such a population behave asymptotically as a single individual whose survival probability from one time step to the next would be λ, the
asymptotic growth rate of the population. In the White Stork case, the high value
of 1 − λ tells us that asymptotically the population will become extinct relatively
quickly.
The aforementioned interplay between extinction and “quasi-stabilization” is
valid as soon as quasi-stationarity occurs, including for DD BPs. The above
parallel between deterministic models and BPs is, however, only valid in densityindependent, constant-environment situations, in which the expected population
sizes of the BPs are given by a Leslie matrix model. For DD BPs, we are not aware
of any simple deterministic model that would yield the successive expected population sizes of the BPs as functions of only the previous expected population size.
Indeed, the interplay between stochasticity and density dependence makes E (Z t )
depend not only on E(Z t−1 ) but on the whole probability distribution of Z t−1 .
Theoretical results also appear useful to frame simulations. We give three
illustrations of this statement:
1. Theoretical results emphasize certain quantities, with known asymptotic behavior, such as the expected population size conditional on nonextinction (e.g.,
Lebreton 1978; Gosselin 1996).
2. They lead to meaningful procedures for estimating some parameters. For example, once quasi-stationarity is reached, mean time to extinction is best
estimated by fitting a truncated geometric distribution to the empirical probability distribution of time to extinction. In simulations over a finite time interval, one may account in this way for the nonobserved tail of the probability
distribution of time to extinction and estimate properly the mean time to
extinction.
3. Last, theoretical results may help to distinguish the transient part of the simulation results (i.e., the part that is influenced by the initial conditions) from the
asymptotic one (e.g., Goodman 1987; Gabriel and B¨ urger 1992). The relative
role of the two components, and thus the relevance of the asymptotic results,
likely depend on a ratio
|λ 2 |
λ
, where λ, the asymptotic rate of growth, is the
largest eigenvalue, in modulus, of an equivalent of the matrix Q introduced in
Appendix 13.2. The second largest eigenvalue is λ 2 . If 1 −
|λ 2 |
λ
is much bigger
than 1 − λ, quasi-stationarity will be reached rapidly, before many extinctions
have occurred, in which case the results presented in this chapter are very
relevant. On the contrary, if 1 −
|λ 2 |
λ
is not much bigger than 1 − λ, many
extinctions are likely to take place before quasi-stationarity is reached. In this
last case, quasi-stationarity is less interesting (cf. Day and Possingham 1995;
Gosselin 1998b, c).